# The Mechanics and Intuition of Potential Energy > Potential energy is usually handed over as a formula to remember. This lecture derives it instead. We start from Newton's second law along the tangent to a particle's path, integrate once, and watch kinetic energy and the work integral fall out of the algebra. Then we face the awkward consequence: if all the work done on a body changes only its kinetic energy, where does potential energy live? The answer is the gradient theorem for line integrals, which makes the work of a conservative force depend on its endpoints alone, and from that single idea both m g h and one half k x squared are derived rather than asserted. We then assemble the general work-energy equation, define power and mechanical efficiency, and finish by solving a spring-and-two-blocks separation problem that is long with F = m a and short with energy. - Canonical watch page: [The Mechanics and Intuition of Potential Energy](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Engineering - Published: 2026-09-07T00:35:05.016Z - Updated: 2026-09-07T00:35:05.016Z - Duration: PT931S (15 minutes 31 seconds) - Chapters: 4 - Views: 2 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M1WJH2B9SWX97THE0DZN415Q/0/dark/master.m3u8) - Embed: [Player](https://academa.ai/embed/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M1WJH2B9SWX97THE0DZN415Q/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M1WJH2B9SWX97THE0DZN415Q/0/dark/poster.jpg) ## Description Work, kinetic energy and potential energy derived from F = ma: the gradient theorem, mgh, spring potential, and a spring-block example. ## Chapters - [00:00–03:47.289 · Work and Kinetic Energy](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=0) - [03:47.289–07:40.063 · Where Potential Energy Comes From](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=227.2892083333333) - [07:40.063–10:23.747 · The Master Equation](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=460.0625) - [10:23.747–15:31 · The Spring and the Two Blocks](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=623.7468124999999) ## Transcript ### [00:00 · Work and Kinetic Energy](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=0) This is the equation we are heading for. Once you have it, a large class of mechanics problems stops needing an integration in time and becomes ordinary algebra. Kinetic energy on the left, potential energy beside it, the work of everything else in the middle, and the same two quantities again at the finish. Not one of those symbols is a new law of physics. They all come out of Newton's second law, and in this chapter we pull them out of it. So, a particle, and the curve it happens to be travelling along. Nothing about that route is special: it bends where it likes, and the particle sits somewhere on it. Several forces act on it, and they change from instant to instant. Right now one of them pushes it forward, another drags it back, and the green arrow is its velocity, which always lies along the curve. Add the forces up and split the resultant into two pieces. The piece across the curve bends the trajectory and does nothing to the speed. Only the piece along it, the tangential one, can make the particle go faster or slower. Let it move, and notice how little stays put. Both forces swing round, the tangential piece changes size, and the speed changes with it. Attacking this by integrating the acceleration directly would be grim. So here is the one piece of kinematics we need. Tangential acceleration is the rate of change of speed with time, and by the chain rule that is also v times the rate of change of speed with distance along the path. Newton's second law along the tangent then reads: the sum of the tangential force components equals m times that acceleration, which is now m v, times d v by d s. Separate the differentials, and the tangential force times a little piece of path sits on one side, m v d v on the other. Now integrate both sides over the whole journey, from the starting point to the finishing point. The right-hand side is an ordinary integral in v alone, so it simply evaluates: one half m v final squared, minus one half m v initial squared. And there is our old friend, arriving unannounced: one half m v squared. Nobody defined it for us. It fell out of an integral, and that is the honest origin of kinetic energy. Give that combination a name: T, the kinetic energy. Our result then says something about the integral on the left. Whatever that integral is, it equals the change in T. The integral deserves a name too. As written it demands the tangential component at every single point, which is a nuisance. But a dot product does that job by itself: dotting the force with the small displacement keeps the part along the motion and throws the rest away. So we define the work done by a force between two points as the line integral of force dotted with displacement, and the theorem reads: the work done on a particle equals the change in its kinetic energy. That is the whole of it. If the force is constant and the path is straight, the integral collapses to force times distance, which is the version most of us met at school. That version is the special case. This one is the rule. Which leaves one uncomfortable question. If all the work done on a body goes into kinetic energy and nowhere else, then where does potential energy live? Lift a book onto a shelf and you have certainly given it something. That is the next chapter. ### [03:47.289 · Where Potential Energy Comes From](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=227.2892083333333) Here is the question the last chapter left us with. All the work done on a body shows up as kinetic energy, and yet everybody talks about energy stored in a raised weight or a squeezed spring. Reconciling those two statements takes no new physics at all. It takes calculus. Some work integrals are far easier than others. Take the simplest force there is: weight, pointing steadily down, the same everywhere. Here it is, and here are two points to travel between. Carry the particle from the first to the second along this high road. Gravity does some amount of work on the way, and computing it from the definition would mean grinding through a line integral. Now the same trip along a completely different route, dipping low. The length is different, the direction of travel is different at every moment, and yet the work done by gravity is exactly the same number. Because the only thing gravity ever cared about was the drop in height. Not the route. Just that one vertical distance. Here is the general statement. A force is called conservative when it can be written as minus the gradient of some scalar function V. Weight can be. Friction cannot. And for such a force, the fundamental theorem for line integrals does the integral for you. The work from i to f is V at the start, minus V at the end. No integration, no route, just two evaluations. So the work done by a conservative force is minus the change in V, and that function V is what we call potential energy. The minus sign is doing real work: when the force does positive work, the potential comes down. In particle dynamics there are exactly two conservative forces to worry about: weight, and the linear spring. Let us build both potentials from that one definition. Weight first. Here is the floor, and here is a block sitting a height y above it. We agree to call the floor the level where the potential is zero. The only force doing work is the block's own weight. Try the function V g equals m g y, and check it against the definition. Its gradient has only one surviving component, and minus that component is minus m g: precisely the weight vector, pointing down. So weight is conservative, and this is its potential. Now the work done by weight as the block falls. Potential at the start minus potential at the end is m g y initial minus m g y final, which is m g times the height given up. So m g h was never a definition. It is the value of a line integral we no longer have to do, and its shape is fixed entirely by the fact that weight is constant. The other conservative force is the linear spring. Rather than draw the spring, draw its potential: one half k x squared, a parabola, with x measured from the natural length. Sit the block at some compression, here. The force is minus the slope of this curve, and the slope of a parabola grows in proportion to x, so the force is minus k x, pointing back toward the middle. Watch the slope as the compression changes. Steeper further out, gentler closer in, and it would vanish altogether at the natural length. That is exactly what Hooke's law says about a spring. Now the work the spring does as the block is released from a compression d and returns to the natural length. Potential at the start, minus potential at the end: one half k d squared, all of it delivered to the block. That is the whole of potential energy. It is a device for computing the work of the forces whose work does not depend on the route. Friction is not one of them, which is why friction is about to get a term of its own. ### [07:40.063 · The Master Equation](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=460.0625) Here is the picture nearly every energy problem reduces to. A rough slope, a block on it, weight pulling down, friction rubbing backwards, and a speed that rises as the height falls. Let it slide. Nothing exotic is happening: height is being traded for speed, and friction is quietly taking a cut of the total. Two facts, one from each of the last two chapters. First, the total work done on the body equals the change in its kinetic energy. Second, that total work splits in two. The conservative forces contribute V initial minus V final, and everything else contributes a term we write U prime: friction, a cable, a hand pushing. Both lines describe the same total work, so set them equal to one another. And then move the potentials to the sides they belong on. Initial kinetic, plus initial potential, plus the work of everything else, equals final kinetic plus final potential. That is the equation from the first minute of this lecture, and now every symbol in it has been earned. Read it on the slope: V is the height term, T is the speed term, and U prime is the friction, always negative, because friction always opposes the motion. Nothing here assumed the friction was small, or the path straight, or the forces constant. This is the general form. And if nothing but weight and springs do any work, U prime is zero and T plus V is the same at both ends, which is conservation of mechanical energy as a special case. Two more definitions finish the vocabulary. Power is the rate at which work is being done: how much per second, rather than how much in total. For a machine that is usually what you care about. An engine that can do a great deal of work per second is a powerful engine, and force dotted with velocity is often the quickest way to get at it. Efficiency is the other one. You pour power into a machine, some of it comes out as useful work, and the rest becomes heat, noise and wear. Efficiency is the ratio of the two, and it is always less than one. The same ratio can be taken over a whole job rather than instant by instant, with energies instead of powers. Fill a tank with fuel, and ask how much of it came back as useful work. That is the entire theory: one master equation, and two bookkeeping definitions. Now watch what it does to a problem that is genuinely painful with Newton's laws. ### [10:23.747 · The Spring and the Two Blocks](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=623.7468124999999) This problem appeared in the chapter on Newton's laws, where it took two pages. Two blocks on a rough floor, one attached to a spring and one merely leaning against it. Push the pair back a distance d, let go, and ask how big d has to be for the second block to leave the first behind. Here is the arrangement. Wall on the left, a floor with kinetic friction coefficient mu k, the spring, block A glued to its end, and block B just touching A. The dashed line marks the natural length, where x is zero. Push the pair back until the spring is compressed by d, and hold them there. Nothing is moving, so the kinetic energy is nothing, and the spring is holding one half k d squared. Now let go. The spring pushes both blocks forward, friction rubs backwards on both, and the pair speeds up until the spring reaches its natural length. Right there the spring stops pushing. And that is the moment of separation, for a reason worth pausing on. Past this line the spring is stretched, so it starts pulling A back. It can pull A, which is attached to it. It cannot pull B, which is only in contact. So from that instant A slows, turns round and comes back, while B carries on with whatever speed it already had. They part company. Which collapses the whole question to a single number: the shared speed at the natural length. If that speed is greater than zero, B separates and keeps going. If it is zero, nothing separates at all. And that is exactly the sort of question energy answers well: an initial state, a final state, and no interest whatsoever in what happened between them. With Newton's laws you would have to solve a differential equation for a force that changes with position, and only then evaluate it. Put the blocks back where they started, compressed by d and stationary, and write down the master equation. Then take its five terms one at a time. Initial kinetic energy first. The blocks are being held at rest, so T initial is zero. Initial potential next. Gravity contributes nothing, because nothing changes height on a level floor. The spring contributes one half k d squared, the result we derived a moment ago. Final potential. At the natural length the spring is undeformed, so V final is zero as well. And final kinetic energy. At that instant the blocks are still moving together with one shared speed, so it is one half the total mass, times that speed squared. Which leaves friction, and friction is not conservative: there is no potential for it, so we compute its work directly. The floor has to hold up both blocks, so the normal force is the total weight, m A plus m B, times g. Kinetic friction is mu k times that normal force, and it points leftwards the whole way, because the blocks are travelling rightwards. Force against motion means negative work. It is minus the friction force times the distance travelled, which is minus mu k, times the total mass, times g d. Notice that this is the one place where energy saves us nothing. For a non-conservative force you always do the work integral yourself. It is simply that this particular integral is trivial: a constant force over a straight run. Now put the five pieces into the master equation, and double both sides to clear the halves. Zero, plus k d squared, minus twice mu k times the total weight times d, equals the total mass times the shared speed squared. Solve for the speed squared. Divide both sides by the total mass, and there it is: k d squared, minus twice mu k times total mass times g d, all over the total mass. B separates only if that speed is genuinely positive. The denominator is positive already, so the whole condition sits in the numerator: k d squared has to beat twice mu k, total mass, g, d. Every compression is positive, so divide one factor of d out of both sides. The answer: d must be bigger than twice mu k, total mass, g, over k. Compare that with the route through Newton's laws. There you write F equals m a for the pair, with a spring force that changes with position, integrate to get speed against position, and only then set the speed to zero. Same answer, several times the work. And that is the shape of every energy problem. Pick the two states, write T plus V at each of them, add the work of anything non-conservative, and solve. Almost all of the mechanics is in choosing the two states well. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M1WJH2B9SWX97THE0DZN415Q/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Work and Kinetic Energy](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=0) Span: 00:00–03:47.289 (0s–227.2892083333333s). #### Objects - across: a Vector \[magenta\] labelled "sum F\_n" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - ((0.8 \* (2.21 \* cos((2.6 \* s)))) / sqrt(…) - drag: a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…) - field: a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)) - head\_goal: a Heading that says "Where This Is Going" - head\_path: a Heading that says "One Particle, Many Forces" - head\_work: a Heading that says "Work, and What It Changes" - particle: a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))) - path: a ParametricCurve \[blue\] drawn in field (function=\) - promise: a Text \[text\] that says "Every symbol in it will be built out of $sum arrow(F) = m arrow(a)$ and nothing else." - push: a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …) - s: a VariableNumber (initial\_value=0.16) - simple: a Math \[text\] that says "$U = F d quad upright("(constant force, straight path)")$" - t\_def: a Math \[text\] that says "$T = frac(1, 2) m v^2$" - tangential: a Vector \[yellow\] labelled "sum F\_t" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (3.5 / sqrt((25.0 + ((2.21 \* cos((2.6 \* …) - target: a Math \[text\] that says "$T\_i + V\_i + U'\_(i arrow.r f) = T\_f + V\_f$" - theorem: a Math \[text\] that says "$U\_(i arrow.r f) = T\_f - T\_i = Delta T$" - velocity: a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…) - work\_def: a Math \[text\] that says "$U\_(i arrow.r f) = integral\_(arrow(r)\_i)^(arrow(r)\_f) arrow(F) dot dif arrow(r)$" - work\_out: a Derivation \[text\] that says "$a\_t &= frac(dif v, dif t) = v frac(dif v, dif s) \\ sum F\_t &= m a\_t = m v frac(dif v, dif s) \\ sum F\_t thin dif s &= m v thin dif v \\ integral\_(s\_i)^(s\_f) sum F\_t thin dif s &= integral\_(v\_i)^(v\_f) m v thin dif v \\ &= frac(1, 2) m v\_f^2 - …$" #### Beats ##### [00:00](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=0) Narration: This is the equation we are heading for. Once you have it, a large class of mechanics problems stops needing an integration in time and becomes ordinary algebra. Board: Empty. Actions: - [00:00](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=0): head\_goal is shown on the screen, written out. - [00:0.522](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=0.522): target is shown on the screen, written out. ##### [00:10.84](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=10.84) Narration: Kinetic energy on the left, potential energy beside it, the work of everything else in the middle, and the same two quantities again at the finish. Not one of those symbols is a new law of physics. They all come out of Newton's second law, and in this chapter we pull them out of it. Board: target — a Math \[text\] that says "$T\_i + V\_i + U'\_(i arrow.r f) = T\_f + V\_f$"; head\_goal — a Heading that says "Where This Is Going" Actions: - [00:11.188](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=11.187999999999999): target (the "T\_i" part) is emphasized. - [00:12.907](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=12.907): target (the "T\_i" part) is no longer emphasized. - [00:12.907](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=12.907): target (the "V\_i" part) is emphasized. - [00:16.158](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=16.158): target (the "U'\_(i arrow.r f)" part) is emphasized. - [00:16.158](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=16.158): target (the "V\_i" part) is no longer emphasized. - [00:21.556](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=21.555999999999997): promise is shown on the screen, written out. - [00:25.585](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=25.585): target (the "U'\_(i arrow.r f)" part) is no longer emphasized. - [00:27.373](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=27.373): head\_goal is hidden from the screen — left the board. - [00:27.373](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=27.373): promise is hidden from the screen — left the board. - [00:27.373](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=27.373): target is hidden from the screen — left the board. ##### [00:28.573](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=28.573) Narration: So, a particle, and the curve it happens to be travelling along. Nothing about that route is special: it bends where it likes, and the particle sits somewhere on it. Board: Empty. Actions: - [00:28.573](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=28.573): head\_path is shown on the screen, written out. - [00:28.573](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=28.573): field is shown on the screen, written out. - [00:30.894](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=30.894000000000002): path is shown on the screen, drawn. - [00:38.336](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=38.336): particle is shown on the screen, written out. ##### [00:40.191](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=40.1905) Narration: Several forces act on it, and they change from instant to instant. Right now one of them pushes it forward, another drags it back, and the green arrow is its velocity, which always lies along the curve. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); head\_path — a Heading that says "One Particle, Many Forces"; path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))) Actions: - [00:45.96](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=45.96): push is shown on the screen, written out. - [00:47.458](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=47.458): drag is shown on the screen, written out. - [00:48.886](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=48.885999999999996): velocity is shown on the screen, written out. ##### [00:53.189](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=53.189499999999995) Narration: Add the forces up and split the resultant into two pieces. The piece across the curve bends the trajectory and does nothing to the speed. Only the piece along it, the tangential one, can make the particle go faster or slower. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); head\_path — a Heading that says "One Particle, Many Forces"; path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))); push — a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …); drag — a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…); velocity — a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…) Actions: - [00:57.915](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=57.91499999999999): across is shown on the screen, written out. - [01:3.58](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=63.57999999999999): tangential is shown on the screen, written out. - [01:7.551](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=67.55099999999999): across is hidden from the screen. ##### [01:8.151](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=68.151) Narration: Let it move, and notice how little stays put. Both forces swing round, the tangential piece changes size, and the speed changes with it. Attacking this by integrating the acceleration directly would be grim. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); head\_path — a Heading that says "One Particle, Many Forces"; path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))); push — a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …); drag — a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…); velocity — a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…); tangential — a Vector \[yellow\] labelled "sum F\_t" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (3.5 / sqrt((25.0 + ((2.21 \* cos((2.6 \* …) Actions: - [01:8.801](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=68.80099999999999): particle is redrawn as the numbers it depends on change. - [01:8.801](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=68.80099999999999): push is redrawn as the numbers it depends on change. - [01:8.801](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=68.80099999999999): drag is redrawn as the numbers it depends on change. - [01:8.801](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=68.80099999999999): velocity is redrawn as the numbers it depends on change. - [01:8.801](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=68.80099999999999): tangential is redrawn as the numbers it depends on change. - [01:8.801](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=68.80099999999999): s ticks to 0.78. ##### [01:23.216](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=83.2165) Narration: So here is the one piece of kinematics we need. Tangential acceleration is the rate of change of speed with time, and by the chain rule that is also v times the rate of change of speed with distance along the path. Board: Unchanged from the preceding beat in this scene. Actions: - [01:31.344](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=91.344): field moves to a new place on the board. - [01:31.344](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=91.344): work\_out is shown on the screen, written out. ##### [01:36.971](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=96.9705) Narration: Newton's second law along the tangent then reads: the sum of the tangential force components equals m times that acceleration, which is now m v, times d v by d s. Board: Unchanged from the preceding beat in this scene. Actions: - [01:39.92](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=99.91999999999999): work\_out is shown on the screen, written out. ##### [01:49.68](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=109.68) Narration: Separate the differentials, and the tangential force times a little piece of path sits on one side, m v d v on the other. Now integrate both sides over the whole journey, from the starting point to the finishing point. Board: Unchanged from the preceding beat in this scene. Actions: - [01:50.028](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=110.02799999999999): work\_out is shown on the screen, written out. - [01:58.504](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=118.50399999999999): work\_out is shown on the screen, written out. ##### [02:3.655](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=123.65450000000001) Narration: The right-hand side is an ordinary integral in v alone, so it simply evaluates: one half m v final squared, minus one half m v initial squared. Board: Unchanged from the preceding beat in this scene. Actions: - [02:8.334](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=128.334): work\_out is shown on the screen, written out. ##### [02:15.54](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=135.54000000000002) Narration: And there is our old friend, arriving unannounced: one half m v squared. Nobody defined it for us. It fell out of an integral, and that is the honest origin of kinetic energy. Board: Unchanged from the preceding beat in this scene. Actions: - [02:18.395](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=138.395): work\_out (the "frac(1, 2) m v\_f^2" part) is emphasized. - [02:27.66](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=147.66): work\_out (the "frac(1, 2) m v\_f^2" part) is no longer emphasized. - [02:29.715](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=149.71500000000003): head\_path is hidden from the screen — left the board. - [02:29.715](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=149.71500000000003): work\_out is hidden from the screen — left the board. ##### [02:30.915](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=150.91500000000002) Narration: Give that combination a name: T, the kinetic energy. Our result then says something about the integral on the left. Whatever that integral is, it equals the change in T. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))); push — a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …); drag — a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…); velocity — a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…); tangential — a Vector \[yellow\] labelled "sum F\_t" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (3.5 / sqrt((25.0 + ((2.21 \* cos((2.6 \* …) Actions: - [02:30.915](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=150.91500000000002): head\_work is shown on the screen, written out. - [02:32.587](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=152.587): t\_def is shown on the screen, written out. ##### [02:44.193](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=164.193) Narration: The integral deserves a name too. As written it demands the tangential component at every single point, which is a nuisance. But a dot product does that job by itself: dotting the force with the small displacement keeps the part along the motion and throws the rest away. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))); push — a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …); drag — a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…); velocity — a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…); tangential — a Vector \[yellow\] labelled "sum F\_t" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (3.5 / sqrt((25.0 + ((2.21 \* cos((2.6 \* …); t\_def — a Math \[text\] that says "$T = frac(1, 2) m v^2$"; head\_work — a Heading that says "Work, and What It Changes" Actions: - [02:51.554](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=171.55399999999997): tangential is indicated — a transient flash. - [02:53.609](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=173.60899999999998): work\_def is shown on the screen, written out. ##### [03:2.371](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=182.371) Narration: So we define the work done by a force between two points as the line integral of force dotted with displacement, and the theorem reads: the work done on a particle equals the change in its kinetic energy. That is the whole of it. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))); push — a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …); drag — a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…); velocity — a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…); tangential — a Vector \[yellow\] labelled "sum F\_t" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (3.5 / sqrt((25.0 + ((2.21 \* cos((2.6 \* …); t\_def — a Math \[text\] that says "$T = frac(1, 2) m v^2$"; work\_def — a Math \[text\] that says "$U\_(i arrow.r f) = integral\_(arrow(r)\_i)^(arrow(r)\_f) arrow(F) dot dif arrow(r)$"; head\_work — a Heading that says "Work, and What It Changes" Actions: - [03:9.569](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=189.56899999999993): theorem is shown on the screen, written out. - [03:16.001](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=196.00099999999995): A box is drawn around theorem. ##### [03:17.565](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=197.565) Narration: If the force is constant and the path is straight, the integral collapses to force times distance, which is the version most of us met at school. That version is the special case. This one is the rule. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))); push — a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …); drag — a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…); velocity — a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…); tangential — a Vector \[yellow\] labelled "sum F\_t" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (3.5 / sqrt((25.0 + ((2.21 \* cos((2.6 \* …); t\_def — a Math \[text\] that says "$T = frac(1, 2) m v^2$"; work\_def — a Math \[text\] that says "$U\_(i arrow.r f) = integral\_(arrow(r)\_i)^(arrow(r)\_f) arrow(F) dot dif arrow(r)$"; theorem — a Math \[text\] that says "$U\_(i arrow.r f) = T\_f - T\_i = Delta T$"; head\_work — a Heading that says "Work, and What It Changes" Actions: - [03:21.501](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=201.50099999999995): simple is shown on the screen, written out. ##### [03:31.156](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=211.1565) Narration: Which leaves one uncomfortable question. If all the work done on a body goes into kinetic energy and nowhere else, then where does potential energy live? Lift a book onto a shelf and you have certainly given it something. That is the next chapter. Board: field — a Figure (x\_range=(0.0, 6.2), y\_range=(0.0, 3.8), aspect=(6.2, 3.8)); path — a ParametricCurve \[blue\] drawn in field (function=\); particle — a Point \[text\] labelled "P" drawn in field (location=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s)))))); push — a Vector \[red\] labelled "arrow(F)\_1" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + 0.45), ((1.6 + (0.85 \* sin((2.6 \* s)))) …); drag — a Vector \[red\] labelled "arrow(F)\_2" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) - 0.9), ((1.6 + (0.85 \* sin((2.6 \* s)))) -…); velocity — a Vector \[green\] labelled "arrow(v)" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (6.75 / sqrt((25.0 + ((2.21 \* cos((2.6 \*…); tangential — a Vector \[yellow\] labelled "sum F\_t" drawn in field (start=((0.55 + (5.0 \* s)), (1.6 + (0.85 \* sin((2.6 \* s))))), end=(((0.55 + (5.0 \* s)) + (3.5 / sqrt((25.0 + ((2.21 \* cos((2.6 \* …); t\_def — a Math \[text\] that says "$T = frac(1, 2) m v^2$"; work\_def — a Math \[text\] that says "$U\_(i arrow.r f) = integral\_(arrow(r)\_i)^(arrow(r)\_f) arrow(F) dot dif arrow(r)$"; theorem — a Math \[text\] that says "$U\_(i arrow.r f) = T\_f - T\_i = Delta T$"; simple — a Math \[text\] that says "$U = F d quad upright("(constant force, straight path)")$"; head\_work — a Heading that says "Work, and What It Changes" Actions: - [03:37.229](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=217.22899999999993): theorem is indicated — a transient flash. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): field is hidden from the screen — left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): path is hidden from the screen — field left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): particle is hidden from the screen — field left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): push is hidden from the screen — field left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): drag is hidden from the screen — field left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): velocity is hidden from the screen — field left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): tangential is hidden from the screen — field left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): head\_work is hidden from the screen — left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): simple is hidden from the screen — left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): t\_def is hidden from the screen — left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): theorem is hidden from the screen — left the board. - [03:46.248](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=226.24754166666662): work\_def is hidden from the screen — left the board. ### Scene 2: [Where Potential Energy Comes From](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=227.2892083333333) Span: 03:47.289–07:40.063 (227.2892083333333s–460.0625s). #### Objects - block: a Polygon \[blue\] labelled "m" drawn in lift (vertices=((1.15, \), (1.9, \, x\_range=(-1.55, 1.55)) - consequence: a Math \[text\] that says "$U\_(i arrow.r f) = - Delta V$" - datum: a Math \[gray\] that says "$V\_g = 0$" drawn in lift - end\_pt: a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)) - fall: a Line \[yellow\] labelled "h" drawn in land (start=(4.6, 0.85), end=(4.6, 3.05), dashed=True) - ftc: a Math \[text\] that says "$integral\_i^f arrow(F) dot dif arrow(r) = V\_i - V\_f$" - grad\_def: a Math \[text\] that says "$arrow(F) = - nabla V$" - grav\_work: a Derivation \[text\] that says "$V\_g &= m g y \\ - frac(partial V\_g, partial y) &= - m g \\ U\_(i arrow.r f) &= V\_i - V\_f = m g y\_i - m g y\_f \\ &= m g h$" - gravity: a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit') - ground: a Line \[gray\] drawn in lift (start=(0.2, 0.35), end=(3.0, 0.35)) - head\_grav: a Heading that says "The Potential of Weight" - head\_spring: a Heading that says "The Potential of a Spring" - height: a Line \[yellow\] labelled "y" drawn in lift (start=(0.7, 0.35), end=(0.7, \), dashed=True) - high: a ParametricCurve \[blue\] labelled "upright("high road")" drawn in land (function=\) - high\_t: a VariableNumber (initial\_value=0.02) - land: a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)) - lift: a Figure (x\_range=(0.0, 3.2), y\_range=(0.0, 3.4), aspect=(3.2, 3.4)) - low: a ParametricCurve \[magenta\] labelled "upright("low road")" drawn in land (function=\) - low\_t: a VariableNumber (initial\_value=0.02) - note: a Text \[text\] that says "Weight and the spring are conservative. Friction is not: its work depends on the route." - pull: a Vector \[red\] labelled "m arrow(g)" drawn in lift (start=(1.5, \), end=(1.5, (y\_var - 0.6))) - question: a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?" - rider\_high: a Point \[green\] drawn in land (location=((0.7 + (3.9 \* high\_t)), ((3.05 - (2.2 \* high\_t)) + (0.5 \* sin(…) - rider\_low: a Point \[green\] drawn in land (location=((0.7 + (3.9 \* low\_t)), ((3.05 - (2.2 \* low\_t)) - (0.85 \* sin((…) - sitting: a PlotPoint \[green\] labelled "x" drawn in well (target='bowl', x=\) - slope\_line: a TangentLine \[yellow\] drawn in well (target='bowl', x=\, length=0.7) - spring\_force: a Vector \[red\] labelled "F = - k x" drawn in well (start=(\, 0.3), end=((stretch - 0.7), 0.3)) - spring\_work: a Derivation \[text\] that says "$V\_e &= frac(1, 2) k x^2 \\ - frac(dif V\_e, dif x) &= - k x \\ U\_(i arrow.r f) &= frac(1, 2) k x\_i^2 - frac(1, 2) k x\_f^2 \\ &= frac(1, 2) k d^2$" - start\_pt: a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)) - stretch: a VariableNumber (initial\_value=1.25) - well: an Axes (x\_range=(-1.7, 1.7), y\_range=(0.0, 3.4), x\_ticks\_every=0.5) - y\_var: a VariableNumber (initial\_value=2.2) #### Beats ##### [03:47.289](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=227.2892083333333) Narration: Here is the question the last chapter left us with. All the work done on a body shows up as kinetic energy, and yet everybody talks about energy stored in a raised weight or a squeezed spring. Reconciling those two statements takes no new physics at all. It takes calculus. Board: Empty. Actions: - [03:47.289](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=227.2892083333333): question is shown on the screen, written out. - [04:4.263](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=244.2632083333333): question moves to a new place on the board. ##### [04:5.463](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=245.4632083333333) Narration: Some work integrals are far easier than others. Take the simplest force there is: weight, pointing steadily down, the same everywhere. Here it is, and here are two points to travel between. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?" Actions: - [04:5.463](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=245.4632083333333): land is shown on the screen, written out. - [04:11.407](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=251.4072083333333): gravity is shown on the screen, written out. - [04:16.423](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=256.4232083333333): start\_pt is shown on the screen, written out. - [04:16.714](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=256.7139191423209): end\_pt is shown on the screen, written out. ##### [04:18.823](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=258.8227083333333) Narration: Carry the particle from the first to the second along this high road. Gravity does some amount of work on the way, and computing it from the definition would mean grinding through a line integral. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?"; land — a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)); gravity — a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit'); start\_pt — a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)); end\_pt — a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)) Actions: - [04:21.667](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=261.6672083333333): high is shown on the screen, drawn. - [04:21.667](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=261.6672083333333): rider\_high is shown on the screen, written out. - [04:22.979](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=262.9792083333333): rider\_high is redrawn as the numbers it depends on change. - [04:22.979](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=262.9792083333333): high\_t ticks to 0.98. ##### [04:29.477](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=269.4772083333333) Narration: Now the same trip along a completely different route, dipping low. The length is different, the direction of travel is different at every moment, and yet the work done by gravity is exactly the same number. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?"; land — a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)); gravity — a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit'); start\_pt — a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)); end\_pt — a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)); high — a ParametricCurve \[blue\] labelled "upright("high road")" drawn in land (function=\); rider\_high — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* high\_t)), ((3.05 - (2.2 \* high\_t)) + (0.5 \* sin(…) Actions: - [04:31.88](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=271.8802083333333): low is shown on the screen, drawn. - [04:31.88](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=271.8802083333333): rider\_low is shown on the screen, written out. - [04:34.226](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=274.2262083333333): rider\_low is redrawn as the numbers it depends on change. - [04:34.226](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=274.2262083333333): low\_t ticks to 0.98. ##### [04:41.56](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=281.55970833333333) Narration: Because the only thing gravity ever cared about was the drop in height. Not the route. Just that one vertical distance. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?"; land — a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)); gravity — a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit'); start\_pt — a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)); end\_pt — a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)); high — a ParametricCurve \[blue\] labelled "upright("high road")" drawn in land (function=\); rider\_high — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* high\_t)), ((3.05 - (2.2 \* high\_t)) + (0.5 \* sin(…); low — a ParametricCurve \[magenta\] labelled "upright("low road")" drawn in land (function=\); rider\_low — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* low\_t)), ((3.05 - (2.2 \* low\_t)) - (0.85 \* sin((…) Actions: - [04:44.044](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=284.0442083333333): fall is shown on the screen, written out. - [04:47.481](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=287.4812083333333): fall is indicated — a transient flash. ##### [04:49.37](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=289.3697083333333) Narration: Here is the general statement. A force is called conservative when it can be written as minus the gradient of some scalar function V. Weight can be. Friction cannot. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?"; land — a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)); gravity — a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit'); start\_pt — a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)); end\_pt — a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)); high — a ParametricCurve \[blue\] labelled "upright("high road")" drawn in land (function=\); rider\_high — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* high\_t)), ((3.05 - (2.2 \* high\_t)) + (0.5 \* sin(…); low — a ParametricCurve \[magenta\] labelled "upright("low road")" drawn in land (function=\); rider\_low — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* low\_t)), ((3.05 - (2.2 \* low\_t)) - (0.85 \* sin((…); fall — a Line \[yellow\] labelled "h" drawn in land (start=(4.6, 0.85), end=(4.6, 3.05), dashed=True) Actions: - [04:55.186](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=295.1862083333333): land moves to a new place on the board. - [04:55.186](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=295.1862083333333): grad\_def is shown on the screen, written out. ##### [05:0.976](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=300.9757083333333) Narration: And for such a force, the fundamental theorem for line integrals does the integral for you. The work from i to f is V at the start, minus V at the end. No integration, no route, just two evaluations. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?"; land — a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)); grad\_def — a Math \[text\] that says "$arrow(F) = - nabla V$"; gravity — a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit'); start\_pt — a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)); end\_pt — a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)); high — a ParametricCurve \[blue\] labelled "upright("high road")" drawn in land (function=\); rider\_high — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* high\_t)), ((3.05 - (2.2 \* high\_t)) + (0.5 \* sin(…); low — a ParametricCurve \[magenta\] labelled "upright("low road")" drawn in land (function=\); rider\_low — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* low\_t)), ((3.05 - (2.2 \* low\_t)) - (0.85 \* sin((…); fall — a Line \[yellow\] labelled "h" drawn in land (start=(4.6, 0.85), end=(4.6, 3.05), dashed=True) Actions: - [05:0.976](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=300.9757083333333): ftc is shown on the screen, written out. - [05:8.057](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=308.0572083333333): ftc (the "V\_i" part) is emphasized. - [05:9.636](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=309.6362083333333): ftc (the "V\_f" part) is emphasized. - [05:9.636](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=309.6362083333333): ftc (the "V\_i" part) is no longer emphasized. - [05:12.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=312.9802083333333): ftc (the "V\_f" part) is no longer emphasized. ##### [05:14.584](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=314.5842083333333) Narration: So the work done by a conservative force is minus the change in V, and that function V is what we call potential energy. The minus sign is doing real work: when the force does positive work, the potential comes down. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?"; land — a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)); grad\_def — a Math \[text\] that says "$arrow(F) = - nabla V$"; ftc — a Math \[text\] that says "$integral\_i^f arrow(F) dot dif arrow(r) = V\_i - V\_f$"; gravity — a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit'); start\_pt — a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)); end\_pt — a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)); high — a ParametricCurve \[blue\] labelled "upright("high road")" drawn in land (function=\); rider\_high — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* high\_t)), ((3.05 - (2.2 \* high\_t)) + (0.5 \* sin(…); low — a ParametricCurve \[magenta\] labelled "upright("low road")" drawn in land (function=\); rider\_low — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* low\_t)), ((3.05 - (2.2 \* low\_t)) - (0.85 \* sin((…); fall — a Line \[yellow\] labelled "h" drawn in land (start=(4.6, 0.85), end=(4.6, 3.05), dashed=True) Actions: - [05:18.619](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=318.6192083333333): consequence is shown on the screen, written out. - [05:21.963](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=321.9632083333333): consequence (the "- Delta V" part) is emphasized. - [05:26.444](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=326.4442083333333): consequence (the "- Delta V" part) is no longer emphasized. ##### [05:27.683](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=327.68270833333327) Narration: In particle dynamics there are exactly two conservative forces to worry about: weight, and the linear spring. Let us build both potentials from that one definition. Weight first. Board: question — a Panel that says "All the work done on a particle changes its kinetic energy. So where does potential energy come from?"; land — a Figure (x\_range=(0.0, 5.4), y\_range=(0.0, 3.6), aspect=(5.4, 3.6)); grad\_def — a Math \[text\] that says "$arrow(F) = - nabla V$"; ftc — a Math \[text\] that says "$integral\_i^f arrow(F) dot dif arrow(r) = V\_i - V\_f$"; consequence — a Math \[text\] that says "$U\_(i arrow.r f) = - Delta V$"; gravity — a VectorField \[gray\] drawn in land (function=\, at=((0.7714285714285715, 0.72), (0.7714285714285715, 1.44), (0.771…, scaling='unit'); start\_pt — a Point \[text\] labelled "i" drawn in land (location=(0.7, 3.05)); end\_pt — a Point \[text\] labelled "f" drawn in land (location=(4.6, 0.85)); high — a ParametricCurve \[blue\] labelled "upright("high road")" drawn in land (function=\); rider\_high — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* high\_t)), ((3.05 - (2.2 \* high\_t)) + (0.5 \* sin(…); low — a ParametricCurve \[magenta\] labelled "upright("low road")" drawn in land (function=\); rider\_low — a Point \[green\] drawn in land (location=((0.7 + (3.9 \* low\_t)), ((3.05 - (2.2 \* low\_t)) - (0.85 \* sin((…); fall — a Line \[yellow\] labelled "h" drawn in land (start=(4.6, 0.85), end=(4.6, 3.05), dashed=True) Actions: - [05:32.373](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=332.3732083333333): fall is indicated — a transient flash. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): consequence is hidden from the screen — left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): ftc is hidden from the screen — left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): grad\_def is hidden from the screen — left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): land is hidden from the screen — left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): gravity is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): start\_pt is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): end\_pt is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): high is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): rider\_high is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): low is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): rider\_low is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): fall is hidden from the screen — land left the board. - [05:39.594](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=339.5942083333333): question is hidden from the screen — left the board. ##### [05:40.794](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=340.7942083333333) Narration: Here is the floor, and here is a block sitting a height y above it. We agree to call the floor the level where the potential is zero. The only force doing work is the block's own weight. Board: Empty. Actions: - [05:40.794](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=340.7942083333333): head\_grav is shown on the screen, written out. - [05:40.794](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=340.7942083333333): lift is shown on the screen, written out. - [05:41.618](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=341.6182083333333): ground is shown on the screen, written out. - [05:42.919](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=342.9192083333333): block is shown on the screen, written out. - [05:43.906](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=343.9062083333333): height is shown on the screen, written out. - [05:48.19](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=348.1902083333333): datum is shown on the screen, written out. - [05:51.487](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=351.48720833333334): pull is shown on the screen, written out. ##### [05:52.831](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=352.8307083333333) Narration: Try the function V g equals m g y, and check it against the definition. Its gradient has only one surviving component, and minus that component is minus m g: precisely the weight vector, pointing down. So weight is conservative, and this is its potential. Board: lift — a Figure (x\_range=(0.0, 3.2), y\_range=(0.0, 3.4), aspect=(3.2, 3.4)); head\_grav — a Heading that says "The Potential of Weight"; ground — a Line \[gray\] drawn in lift (start=(0.2, 0.35), end=(3.0, 0.35)); block — a Polygon \[blue\] labelled "m" drawn in lift (vertices=((1.15, \), (1.9, \), dashed=True); datum — a Math \[gray\] that says "$V\_g = 0$" drawn in lift; pull — a Vector \[red\] labelled "m arrow(g)" drawn in lift (start=(1.5, \), end=(1.5, (y\_var - 0.6))) Actions: - [05:53.597](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=353.5972083333333): lift moves to a new place on the board. - [05:53.597](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=353.5972083333333): grav\_work is shown on the screen, written out. - [05:58.751](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=358.7512083333333): grav\_work is shown on the screen, written out. ##### [06:10.938](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=370.9382083333333) Narration: Now the work done by weight as the block falls. Potential at the start minus potential at the end is m g y initial minus m g y final, which is m g times the height given up. Board: Unchanged from the preceding beat in this scene. Actions: - [06:13.051](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=373.0512083333333): block is redrawn as the numbers it depends on change. - [06:13.051](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=373.0512083333333): height is redrawn as the numbers it depends on change. - [06:13.051](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=373.0512083333333): pull is redrawn as the numbers it depends on change. - [06:13.051](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=373.0512083333333): y\_var ticks to 0.75. - [06:14.224](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=374.2242083333333): grav\_work is shown on the screen, written out. - [06:21.898](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=381.8982083333333): grav\_work is shown on the screen, written out. ##### [06:23.659](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=383.65870833333327) Narration: So m g h was never a definition. It is the value of a line integral we no longer have to do, and its shape is fixed entirely by the fact that weight is constant. Board: Unchanged from the preceding beat in this scene. Actions: - [06:27.2](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=387.2002083333333): A box is drawn around grav\_work. - [06:32.982](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=392.9822083333333): pull is indicated — a transient flash. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): grav\_work is hidden from the screen — left the board. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): head\_grav is hidden from the screen — left the board. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): lift is hidden from the screen — left the board. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): ground is hidden from the screen — lift left the board. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): block is hidden from the screen — lift left the board. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): height is hidden from the screen — lift left the board. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): datum is hidden from the screen — lift left the board. - [06:33.98](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=393.9802083333333): pull is hidden from the screen — lift left the board. ##### [06:34.58](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=394.5802083333333) Narration: The other conservative force is the linear spring. Rather than draw the spring, draw its potential: one half k x squared, a parabola, with x measured from the natural length. Board: Empty. Actions: - [06:34.58](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=394.5802083333333): head\_spring is shown on the screen, written out. - [06:34.58](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=394.5802083333333): well is shown on the screen, written out. - [06:42.324](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=402.32420833333333): well moves to a new place on the board. - [06:42.324](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=402.32420833333333): spring\_work is shown on the screen, written out. - [06:43.218](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=403.21820833333334): bowl is shown on the screen, drawn. ##### [06:47.313](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=407.31270833333326) Narration: Sit the block at some compression, here. The force is minus the slope of this curve, and the slope of a parabola grows in proportion to x, so the force is minus k x, pointing back toward the middle. Board: well — an Axes (x\_range=(-1.7, 1.7), y\_range=(0.0, 3.4), x\_ticks\_every=0.5); head\_spring — a Heading that says "The Potential of a Spring"; bowl — a FunctionPlot \[blue\] labelled "frac(1, 2) k x^2" drawn in well (function=\, x\_range=(-1.55, 1.55)) Actions: - [06:49.495](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=409.49520833333327): sitting is shown on the screen, written out. - [06:51.62](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=411.62020833333327): slope\_line is shown on the screen, written out. - [06:54.127](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=414.12720833333333): spring\_work is shown on the screen, written out. - [06:57.726](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=417.72620833333326): spring\_force is shown on the screen, written out. ##### [07:0.068](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=420.0682083333333) Narration: Watch the slope as the compression changes. Steeper further out, gentler closer in, and it would vanish altogether at the natural length. That is exactly what Hooke's law says about a spring. Board: well — an Axes (x\_range=(-1.7, 1.7), y\_range=(0.0, 3.4), x\_ticks\_every=0.5); head\_spring — a Heading that says "The Potential of a Spring"; bowl — a FunctionPlot \[blue\] labelled "frac(1, 2) k x^2" drawn in well (function=\, x\_range=(-1.55, 1.55)); sitting — a PlotPoint \[green\] labelled "x" drawn in well (target='bowl', x=\); slope\_line — a TangentLine \[yellow\] drawn in well (target='bowl', x=\, length=0.7); spring\_force — a Vector \[red\] labelled "F = - k x" drawn in well (start=(\, 0.3), end=((stretch - 0.7), 0.3)) Actions: - [07:4.503](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=424.5032083333333): sitting is redrawn as the numbers it depends on change. - [07:4.503](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=424.5032083333333): slope\_line is redrawn as the numbers it depends on change. - [07:4.503](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=424.5032083333333): spring\_force is redrawn as the numbers it depends on change. - [07:4.503](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=424.5032083333333): stretch ticks to 0.9. - [07:10.064](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=430.0642083333333): sitting is redrawn as the numbers it depends on change. - [07:10.064](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=430.0642083333333): slope\_line is redrawn as the numbers it depends on change. - [07:10.064](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=430.0642083333333): spring\_force is redrawn as the numbers it depends on change. - [07:10.064](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=430.0642083333333): stretch ticks to 1.4. ##### [07:12.557](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=432.5567083333333) Narration: Now the work the spring does as the block is released from a compression d and returns to the natural length. Potential at the start, minus potential at the end: one half k d squared, all of it delivered to the block. Board: Unchanged from the preceding beat in this scene. Actions: - [07:16.574](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=436.5742083333333): sitting is redrawn as the numbers it depends on change. - [07:16.574](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=436.5742083333333): slope\_line is redrawn as the numbers it depends on change. - [07:16.574](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=436.5742083333333): spring\_force is redrawn as the numbers it depends on change. - [07:16.574](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=436.5742083333333): stretch ticks to 0.4. - [07:18.757](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=438.7572083333333): spring\_work is shown on the screen, written out. - [07:23.18](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=443.1802083333333): spring\_work is shown on the screen, written out. - [07:24.515](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=444.51520833333325): A box is drawn around spring\_work. ##### [07:26.462](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=446.46220833333325) Narration: That is the whole of potential energy. It is a device for computing the work of the forces whose work does not depend on the route. Friction is not one of them, which is why friction is about to get a term of its own. Board: Unchanged from the preceding beat in this scene. Actions: - [07:34.589](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=454.5892083333333): note is shown on the screen, written out. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): head\_spring is hidden from the screen — left the board. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): note is hidden from the screen — left the board. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): spring\_work is hidden from the screen — left the board. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): well is hidden from the screen — left the board. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): bowl is hidden from the screen — well left the board. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): sitting is hidden from the screen — well left the board. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): slope\_line is hidden from the screen — well left the board. - [07:39.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=459.02083333333326): spring\_force is hidden from the screen — well left the board. ### Scene 3: [The Master Equation](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=460.0625) Span: 07:40.063–10:23.747 (460.0625s–623.7468124999999s). #### Objects - bench: a Line \[gray\] drawn in gauge (start=(0.3, 0.25), end=(3.7, 0.25)) - conserved: a Math \[text\] that says "$U'\_(i arrow.r f) = 0 quad arrow.r quad T\_i + V\_i = T\_f + V\_f$" - drop: a Line \[yellow\] labelled "h" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), 0.7), end=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), dashed=True) - eff\_energy: a Math \[text\] that says "$epsilon = frac(U\_upright("out"), U\_upright("in"))$" - eff\_power: a Math \[text\] that says "$epsilon = frac(P\_upright("out"), P\_upright("in")) \<= 1$" - floor: a Line \[gray\] drawn in slope (start=(0.3, 0.7), end=(5.8, 0.7)) - gauge: a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.4), aspect=(4.0, 3.4)) - head\_master: a Heading that says "Putting the Two Facts Together" - head\_power: a Heading that says "Power and Efficiency" - in\_bar: a Polygon \[yellow\] labelled "P\_upright("in")" drawn in gauge (vertices=((0.55, 0.25), (1.55, 0.25), (1.55, 3.0), (0.55, 3.0)), fill\_opacity=0.5) - incline: a Line \[gray\] drawn in slope (start=(0.5, 3.0), end=(5.2, 0.7)) - loss\_bar: a Polygon \[gray\] labelled "upright("losses")" drawn in gauge (vertices=((2.3, 1.85), (3.3, 1.85), (3.3, 3.0), (2.3, 3.0)), fill\_opacity=0.35) - master: a Derivation \[text\] that says "$U\_(i arrow.r f) &= T\_f - T\_i \\ U\_(i arrow.r f) &= (V\_i - V\_f) + U'\_(i arrow.r f) \\ (V\_i - V\_f) + U'\_(i arrow.r f) &= T\_f - T\_i \\ T\_i + V\_i + U'\_(i arrow.r f) &= T\_f + V\_f$" - out\_bar: a Polygon \[green\] labelled "P\_upright("out")" drawn in gauge (vertices=((2.3, 0.25), (3.3, 0.25), (3.3, 1.85), (2.3, 1.85)), fill\_opacity=0.5) - power: a Math \[text\] that says "$P = frac(dif U, dif t) = arrow(F) dot arrow(v)$" - puck: a Circle \[blue\] drawn in slope (center=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), radius=0.22, filled=True) - rub: a Vector \[magenta\] labelled "F\_f" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), end=((((0.5 + (4.7 \* u)) + 0.097) - 0.67), (((3.0 - (2.3 \* u)) + 0.…) - slope: a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.4), aspect=(6.0, 3.4)) - speed: a Vector \[green\] labelled "arrow(v)" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), end=((((0.5 + (4.7 \* u)) + 0.097) + ((0.3 + (0.9 \* u)) \* 0.899)), (…) - u: a VariableNumber (initial\_value=0.1) - weight: a Vector \[red\] labelled "m arrow(g)" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), end=(((0.5 + (4.7 \* u)) + 0.097), (((3.0 - (2.3 \* u)) + 0.198) - 0.…) #### Beats ##### [07:40.063](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=460.0625) Narration: Here is the picture nearly every energy problem reduces to. A rough slope, a block on it, weight pulling down, friction rubbing backwards, and a speed that rises as the height falls. Board: Empty. Actions: - [07:40.063](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=460.0625): head\_master is shown on the screen, written out. - [07:40.063](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=460.0625): slope is shown on the screen, written out. - [07:43.976](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=463.9755): incline is shown on the screen, written out. - [07:43.976](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=463.9755): floor is shown on the screen, written out. - [07:44.776](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=464.7765): puck is shown on the screen, written out. - [07:45.868](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=465.8675): weight is shown on the screen, written out. - [07:47.284](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=467.2835): rub is shown on the screen, written out. - [07:49.084](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=469.0835): speed is shown on the screen, written out. - [07:50.325](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=470.3255): drop is shown on the screen, written out. ##### [07:52.261](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=472.261) Narration: Let it slide. Nothing exotic is happening: height is being traded for speed, and friction is quietly taking a cut of the total. Board: slope — a Figure (x\_range=(0.0, 6.0), y\_range=(0.0, 3.4), aspect=(6.0, 3.4)); head\_master — a Heading that says "Putting the Two Facts Together"; incline — a Line \[gray\] drawn in slope (start=(0.5, 3.0), end=(5.2, 0.7)); floor — a Line \[gray\] drawn in slope (start=(0.3, 0.7), end=(5.8, 0.7)); puck — a Circle \[blue\] drawn in slope (center=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), radius=0.22, filled=True); weight — a Vector \[red\] labelled "m arrow(g)" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), end=(((0.5 + (4.7 \* u)) + 0.097), (((3.0 - (2.3 \* u)) + 0.198) - 0.…); rub — a Vector \[magenta\] labelled "F\_f" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), end=((((0.5 + (4.7 \* u)) + 0.097) - 0.67), (((3.0 - (2.3 \* u)) + 0.…); speed — a Vector \[green\] labelled "arrow(v)" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), end=((((0.5 + (4.7 \* u)) + 0.097) + ((0.3 + (0.9 \* u)) \* 0.899)), (…); drop — a Line \[yellow\] labelled "h" drawn in slope (start=(((0.5 + (4.7 \* u)) + 0.097), 0.7), end=(((0.5 + (4.7 \* u)) + 0.097), ((3.0 - (2.3 \* u)) + 0.198)), dashed=True) Actions: - [07:53.166](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=473.1665): puck is redrawn as the numbers it depends on change. - [07:53.166](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=473.1665): weight is redrawn as the numbers it depends on change. - [07:53.166](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=473.1665): rub is redrawn as the numbers it depends on change. - [07:53.166](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=473.1665): speed is redrawn as the numbers it depends on change. - [07:53.166](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=473.1665): drop is redrawn as the numbers it depends on change. - [07:53.166](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=473.1665): u ticks to 0.82. ##### [08:1.476](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=481.4755) Narration: Two facts, one from each of the last two chapters. First, the total work done on the body equals the change in its kinetic energy. Board: Unchanged from the preceding beat in this scene. Actions: - [08:5.678](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=485.6785): slope moves to a new place on the board. - [08:5.678](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=485.6785): master is shown on the screen, written out. ##### [08:10.748](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=490.748) Narration: Second, that total work splits in two. The conservative forces contribute V initial minus V final, and everything else contributes a term we write U prime: friction, a cable, a hand pushing. Board: Unchanged from the preceding beat in this scene. Actions: - [08:12.663](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=492.6635): master is shown on the screen, written out. - [08:14.416](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=494.4165): weight is indicated — a transient flash. - [08:21.337](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=501.3365): rub is indicated — a transient flash. ##### [08:25.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=505.129) Narration: Both lines describe the same total work, so set them equal to one another. Board: Unchanged from the preceding beat in this scene. Actions: - [08:28.716](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=508.7165): master is shown on the screen, written out. ##### [08:30.814](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=510.814) Narration: And then move the potentials to the sides they belong on. Initial kinetic, plus initial potential, plus the work of everything else, equals final kinetic plus final potential. Board: Unchanged from the preceding beat in this scene. Actions: - [08:31.673](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=511.6735): master is shown on the screen, written out. ##### [08:42.815](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=522.815) Narration: That is the equation from the first minute of this lecture, and now every symbol in it has been earned. Read it on the slope: V is the height term, T is the speed term, and U prime is the friction, always negative, because friction always opposes the motion. Board: Unchanged from the preceding beat in this scene. Actions: - [08:48.157](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=528.1565): A box is drawn around master. - [08:51.929](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=531.9295): master (the "V\_i" part) is emphasized. - [08:53.519](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=533.5195): master (the "T\_i" part) is emphasized. - [08:53.519](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=533.5195): master (the "V\_i" part) is no longer emphasized. - [08:54.832](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=534.8315): master (the "T\_i" part) is no longer emphasized. - [08:54.832](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=534.8315): master (the "U'\_(i arrow.r f)" part) is emphasized. - [08:58.965](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=538.9645): master (the "U'\_(i arrow.r f)" part) is no longer emphasized. ##### [09:1.063](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=541.063) Narration: Nothing here assumed the friction was small, or the path straight, or the forces constant. This is the general form. And if nothing but weight and springs do any work, U prime is zero and T plus V is the same at both ends, which is conservation of mechanical energy as a special case. Board: Unchanged from the preceding beat in this scene. Actions: - [09:16.157](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=556.1565): conserved is shown on the screen, written out. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): conserved is hidden from the screen — left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): head\_master is hidden from the screen — left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): master is hidden from the screen — left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): slope is hidden from the screen — left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): incline is hidden from the screen — slope left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): floor is hidden from the screen — slope left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): puck is hidden from the screen — slope left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): weight is hidden from the screen — slope left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): rub is hidden from the screen — slope left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): speed is hidden from the screen — slope left the board. - [09:19.57](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=559.5695000000001): drop is hidden from the screen — slope left the board. ##### [09:20.769](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=560.7695) Narration: Two more definitions finish the vocabulary. Power is the rate at which work is being done: how much per second, rather than how much in total. Board: Empty. Actions: - [09:20.769](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=560.7695): head\_power is shown on the screen, written out. - [09:20.769](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=560.7695): gauge is shown on the screen, written out. - [09:20.769](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=560.7695): bench is shown on the screen, written out. - [09:25.007](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=565.0074999999999): in\_bar is shown on the screen, written out. - [09:28.014](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=568.0145): gauge moves to a new place on the board. - [09:28.014](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=568.0145): power is shown on the screen, written out. ##### [09:31.226](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=571.226) Narration: For a machine that is usually what you care about. An engine that can do a great deal of work per second is a powerful engine, and force dotted with velocity is often the quickest way to get at it. Board: power — a Math \[text\] that says "$P = frac(dif U, dif t) = arrow(F) dot arrow(v)$"; gauge — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.4), aspect=(4.0, 3.4)); head\_power — a Heading that says "Power and Efficiency"; bench — a Line \[gray\] drawn in gauge (start=(0.3, 0.25), end=(3.7, 0.25)); in\_bar — a Polygon \[yellow\] labelled "P\_upright("in")" drawn in gauge (vertices=((0.55, 0.25), (1.55, 0.25), (1.55, 3.0), (0.55, 3.0)), fill\_opacity=0.5) Actions: - [09:37.611](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=577.6115): in\_bar is indicated — a transient flash. - [09:39.91](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=579.9105): power (the "arrow(F) dot arrow(v)" part) is emphasized. - [09:42.952](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=582.952): power (the "arrow(F) dot arrow(v)" part) is no longer emphasized. ##### [09:43.552](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=583.552) Narration: Efficiency is the other one. You pour power into a machine, some of it comes out as useful work, and the rest becomes heat, noise and wear. Efficiency is the ratio of the two, and it is always less than one. Board: Unchanged from the preceding beat in this scene. Actions: - [09:49.67](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=589.6705): out\_bar is shown on the screen, written out. - [09:51.772](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=591.7715000000001): loss\_bar is shown on the screen, written out. - [09:55.44](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=595.4404999999999): eff\_power is shown on the screen, written out. - [09:57.716](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=597.7155): eff\_power (the "\<= 1" part) is emphasized. - [09:59.086](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=599.086): eff\_power (the "\<= 1" part) is no longer emphasized. ##### [09:59.686](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=599.6859999999999) Narration: The same ratio can be taken over a whole job rather than instant by instant, with energies instead of powers. Fill a tank with fuel, and ask how much of it came back as useful work. Board: power — a Math \[text\] that says "$P = frac(dif U, dif t) = arrow(F) dot arrow(v)$"; eff\_power — a Math \[text\] that says "$epsilon = frac(P\_upright("out"), P\_upright("in")) \<= 1$"; gauge — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.4), aspect=(4.0, 3.4)); head\_power — a Heading that says "Power and Efficiency"; bench — a Line \[gray\] drawn in gauge (start=(0.3, 0.25), end=(3.7, 0.25)); in\_bar — a Polygon \[yellow\] labelled "P\_upright("in")" drawn in gauge (vertices=((0.55, 0.25), (1.55, 0.25), (1.55, 3.0), (0.55, 3.0)), fill\_opacity=0.5); out\_bar — a Polygon \[green\] labelled "P\_upright("out")" drawn in gauge (vertices=((2.3, 0.25), (3.3, 0.25), (3.3, 1.85), (2.3, 1.85)), fill\_opacity=0.5); loss\_bar — a Polygon \[gray\] labelled "upright("losses")" drawn in gauge (vertices=((2.3, 1.85), (3.3, 1.85), (3.3, 3.0), (2.3, 3.0)), fill\_opacity=0.35) Actions: - [10:4.83](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=604.8295): eff\_energy is shown on the screen, written out. - [10:8.149](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=608.1495): loss\_bar is indicated — a transient flash. ##### [10:12.198](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=612.198) Narration: That is the entire theory: one master equation, and two bookkeeping definitions. Now watch what it does to a problem that is genuinely painful with Newton's laws. Board: power — a Math \[text\] that says "$P = frac(dif U, dif t) = arrow(F) dot arrow(v)$"; eff\_power — a Math \[text\] that says "$epsilon = frac(P\_upright("out"), P\_upright("in")) \<= 1$"; eff\_energy — a Math \[text\] that says "$epsilon = frac(U\_upright("out"), U\_upright("in"))$"; gauge — a Figure (x\_range=(0.0, 4.0), y\_range=(0.0, 3.4), aspect=(4.0, 3.4)); head\_power — a Heading that says "Power and Efficiency"; bench — a Line \[gray\] drawn in gauge (start=(0.3, 0.25), end=(3.7, 0.25)); in\_bar — a Polygon \[yellow\] labelled "P\_upright("in")" drawn in gauge (vertices=((0.55, 0.25), (1.55, 0.25), (1.55, 3.0), (0.55, 3.0)), fill\_opacity=0.5); out\_bar — a Polygon \[green\] labelled "P\_upright("out")" drawn in gauge (vertices=((2.3, 0.25), (3.3, 0.25), (3.3, 1.85), (2.3, 1.85)), fill\_opacity=0.5); loss\_bar — a Polygon \[gray\] labelled "upright("losses")" drawn in gauge (vertices=((2.3, 1.85), (3.3, 1.85), (3.3, 3.0), (2.3, 3.0)), fill\_opacity=0.35) Actions: - [10:16.899](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=616.8995): power is indicated — a transient flash. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): eff\_energy is hidden from the screen — left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): eff\_power is hidden from the screen — left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): gauge is hidden from the screen — left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): bench is hidden from the screen — gauge left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): in\_bar is hidden from the screen — gauge left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): out\_bar is hidden from the screen — gauge left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): loss\_bar is hidden from the screen — gauge left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): head\_power is hidden from the screen — left the board. - [10:22.705](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=622.7051458333333): power is hidden from the screen — left the board. ### Scene 4: [The Spring and the Two Blocks](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=623.7468124999999) Span: 10:23.747–15:31.171 (623.7468124999999s–931.1710416666667s). #### Objects - bench: a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)) - block\_a: a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45) - block\_b: a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45) - coil: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)) - coil\_2: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)) - coil\_3: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)) - coil\_4: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)) - coil\_5: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)) - coil\_6: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)) - coil\_7: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)) - coil\_8: a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)) - comp: a VariableNumber - compare: a Block \[text\] that says "Energy: two states, one equation. $F = m a$: integrate first." - d\_mark: a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)) - fric\_a: a Vector \[magenta\] labelled "F\_f" drawn in stage (start=(((0.4 + (2.3 - comp)) + 0.4), 0.78), end=(((0.4 + (2.3 - comp)) - 0.1), 0.78)) - fric\_b: a Vector \[magenta\] drawn in stage (start=(((((0.4 + (2.3 - comp)) + 0.75) + sep) + 0.35), 0.75), end=(((((0.4 + (2.3 - comp)) + 0.75) + sep) - 0.1), 0.75)) - head\_friction: a Heading that says "The Work of Friction" - head\_setup: a Heading that says "The Five Terms" - head\_solve: a Heading that says "Solving for the Condition" - mu\_mark: a Math \[gray\] that says "$mu\_k$" drawn in stage - neutral: a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True) - point: a Point \[yellow\] drawn in stage (location=(1.7499999999999998, 1.35)) - point\_2: a Point \[yellow\] drawn in stage (location=(2.6999999999999997, 1.35)) - point\_3: a Point \[yellow\] drawn in stage (location=(1.7499999999999998, 1.35)) - point\_4: a Point \[yellow\] drawn in stage (location=(2.6999999999999997, 1.35)) - question: a Panel that says "Block $A$ is attached to a spring of stiffness $k$; block $B$ merely rests against $A$ on a floor with friction coefficient $mu\_k$. The pair is pushed back a distance $d$ and released. How large must $d$ be for $B$ to separate from $A$?" - rough: a Derivation \[text\] that says "$N &= (m\_A + m\_B) g \\ F\_f &= mu\_k N = mu\_k (m\_A + m\_B) g \\ U'\_(i arrow.r f) &= - F\_f d \\ &= - mu\_k (m\_A + m\_B) g d$" - sep: a VariableNumber - setup: a Derivation \[text\] that says "$T\_i + V\_i + U'\_(i arrow.r f) &= T\_f + V\_f \\ T\_i &= 0 \\ V\_i &= frac(1, 2) k d^2 \\ V\_f &= 0 \\ T\_f &= frac(1, 2) (m\_A + m\_B) v^2$" - shared\_v: a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)) - solve: a Derivation \[text\] that says "$k d^2 - 2 mu\_k (m\_A + m\_B) g d &= (m\_A + m\_B) v^2 \\ v^2 &= frac(k d^2 - 2 mu\_k (m\_A + m\_B) g d, m\_A + m\_B) \\ k d^2 &\> 2 mu\_k (m\_A + m\_B) g d \\ d &\> frac(2 mu\_k (m\_A + m\_B) g, k)$" - stage: a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)) - state\_f: a Math \[text\] that says "$upright("state") thin f: quad x = 0, quad v = v$" - state\_i: a Math \[text\] that says "$upright("state") thin i: quad x = d, quad v = 0$" - wall: a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)) #### Beats ##### [10:23.747](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=623.7468124999999) Narration: This problem appeared in the chapter on Newton's laws, where it took two pages. Two blocks on a rough floor, one attached to a spring and one merely leaning against it. Push the pair back a distance d, let go, and ask how big d has to be for the second block to leave the first behind. Board: Empty. Actions: - [10:23.747](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=623.7468124999999): question is shown on the screen, written out. - [10:40.918](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=640.9183125): question moves to a new place on the board. ##### [10:42.118](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=642.1183124999999) Narration: Here is the arrangement. Wall on the left, a floor with kinetic friction coefficient mu k, the spring, block A glued to its end, and block B just touching A. The dashed line marks the natural length, where x is zero. Board: question — a Panel that says "Block $A$ is attached to a spring of stiffness $k$; block $B$ merely rests against $A$ on a floor with friction coefficient $mu\_k$. The pair is pushed back a distance $d$ and released. How large must $d$ be for $B$ to separate from $A$?" Actions: - [10:42.118](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=642.1183124999999): stage is shown on the screen, written out. - [10:44.475](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=644.4748124999999): wall is shown on the screen, written out. - [10:45.973](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=645.9728124999999): bench is shown on the screen, written out. - [10:47.424](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=647.4238124999999): mu\_mark is shown on the screen, written out. - [10:49.293](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.2928125): coil is shown on the screen, written out. - [10:49.391](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.3905039056287): coil\_2 is shown on the screen, written out. - [10:49.488](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.4881953112576): coil\_3 is shown on the screen, written out. - [10:49.586](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.5858867168865): coil\_4 is shown on the screen, written out. - [10:49.684](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.6835781225154): coil\_5 is shown on the screen, written out. - [10:49.781](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.7812695281443): coil\_6 is shown on the screen, written out. - [10:49.879](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.8789609337732): coil\_7 is shown on the screen, written out. - [10:49.977](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=649.9766523394021): coil\_8 is shown on the screen, written out. - [10:50.78](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=650.7798124999999): block\_a is shown on the screen, written out. - [10:53.067](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=653.0668125): block\_b is shown on the screen, written out. - [10:54.843](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=654.8428124999999): neutral is shown on the screen, written out. ##### [10:59.333](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.3328124999999) Narration: Push the pair back until the spring is compressed by d, and hold them there. Nothing is moving, so the kinetic energy is nothing, and the spring is holding one half k d squared. Board: question — a Panel that says "Block $A$ is attached to a spring of stiffness $k$; block $B$ merely rests against $A$ on a floor with friction coefficient $mu\_k$. The pair is pushed back a distance $d$ and released. How large must $d$ be for $B$ to separate from $A$?"; stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True) Actions: - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil\_2 is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil\_3 is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil\_4 is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil\_5 is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil\_6 is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil\_7 is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): coil\_8 is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): block\_a is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): block\_b is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): shared\_v is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): fric\_a is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): fric\_b is redrawn as the numbers it depends on change. - [10:59.809](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=659.8088124999999): comp ticks to 0.95. - [11:1.515](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=661.5148125): d\_mark is shown on the screen, written out. - [11:6.508](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=666.5078124999999): stage moves to a new place on the board. - [11:6.508](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=666.5078124999999): state\_i is shown on the screen, written out. ##### [11:11.02](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.0198125) Narration: Now let go. The spring pushes both blocks forward, friction rubs backwards on both, and the pair speeds up until the spring reaches its natural length. Right there the spring stops pushing. Board: question — a Panel that says "Block $A$ is attached to a spring of stiffness $k$; block $B$ merely rests against $A$ on a floor with friction coefficient $mu\_k$. The pair is pushed back a distance $d$ and released. How large must $d$ be for $B$ to separate from $A$?"; stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); state\_i — a Math \[text\] that says "$upright("state") thin i: quad x = d, quad v = 0$"; wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)) Actions: - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil\_2 is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil\_3 is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil\_4 is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil\_5 is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil\_6 is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil\_7 is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): coil\_8 is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): block\_a is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): block\_b is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): fric\_a is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): fric\_b is redrawn as the numbers it depends on change. - [11:11.682](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=671.6818125): comp ticks to 0.0. - [11:17.382](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=677.3818124999999): shared\_v is redrawn as the numbers it depends on change. - [11:17.382](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=677.3818124999999): shared\_v is shown on the screen, written out. ##### [11:23.462](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=683.4623124999999) Narration: And that is the moment of separation, for a reason worth pausing on. Past this line the spring is stretched, so it starts pulling A back. It can pull A, which is attached to it. It cannot pull B, which is only in contact. Board: question — a Panel that says "Block $A$ is attached to a spring of stiffness $k$; block $B$ merely rests against $A$ on a floor with friction coefficient $mu\_k$. The pair is pushed back a distance $d$ and released. How large must $d$ be for $B$ to separate from $A$?"; stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); state\_i — a Math \[text\] that says "$upright("state") thin i: quad x = d, quad v = 0$"; wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)) Actions: - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil\_2 is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil\_3 is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil\_4 is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil\_5 is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil\_6 is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil\_7 is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): coil\_8 is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): block\_a is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): block\_b is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): shared\_v is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): fric\_a is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): fric\_b is redrawn as the numbers it depends on change. - [11:29.975](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=689.9748124999999): comp ticks to -0.28. ##### [11:39.213](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=699.2128124999999) Narration: So from that instant A slows, turns round and comes back, while B carries on with whatever speed it already had. They part company. Board: Unchanged from the preceding beat in this scene. Actions: - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil\_2 is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil\_3 is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil\_4 is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil\_5 is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil\_6 is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil\_7 is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): coil\_8 is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): block\_a is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): block\_b is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): shared\_v is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): fric\_a is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): fric\_b is redrawn as the numbers it depends on change. - [11:41.941](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=701.9408124999999): comp ticks to 0.22. - [11:44.229](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=704.2288124999999): sep ticks to 1.55. ##### [11:49.228](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=709.2283124999999) Narration: Which collapses the whole question to a single number: the shared speed at the natural length. If that speed is greater than zero, B separates and keeps going. If it is zero, nothing separates at all. Board: Unchanged from the preceding beat in this scene. Actions: - [11:52.851](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=712.8508125): state\_f is shown on the screen, written out. - [11:53.896](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=713.8958124999999): neutral is indicated — a transient flash. ##### [12:3.104](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=723.1043124999999) Narration: And that is exactly the sort of question energy answers well: an initial state, a final state, and no interest whatsoever in what happened between them. With Newton's laws you would have to solve a differential equation for a force that changes with position, and only then evaluate it. Board: question — a Panel that says "Block $A$ is attached to a spring of stiffness $k$; block $B$ merely rests against $A$ on a floor with friction coefficient $mu\_k$. The pair is pushed back a distance $d$ and released. How large must $d$ be for $B$ to separate from $A$?"; stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); state\_i — a Math \[text\] that says "$upright("state") thin i: quad x = d, quad v = 0$"; state\_f — a Math \[text\] that says "$upright("state") thin f: quad x = 0, quad v = v$"; wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)) Actions: - [12:6.883](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=726.8828124999999): point is shown on the screen, grown. - [12:8.056](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=728.0558124999999): point\_2 is shown on the screen, grown. - [12:9.217](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=729.2165997741114): point is hidden from the screen. - [12:10.052](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=730.052274815182): point\_2 is hidden from the screen. - [12:19.492](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=739.4923124999999): stage moves to a new place on the board. - [12:19.492](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=739.4923124999999): question is hidden from the screen — left the board. - [12:19.492](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=739.4923124999999): state\_f is hidden from the screen — left the board. - [12:19.492](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=739.4923124999999): state\_i is hidden from the screen — left the board. ##### [12:20.692](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=740.6923125) Narration: Put the blocks back where they started, compressed by d and stationary, and write down the master equation. Then take its five terms one at a time. Board: stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)) Actions: - [12:20.692](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=740.6923125): head\_setup is shown on the screen, written out. - [12:21.099](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=741.0988124999999): block\_b is redrawn as the numbers it depends on change. - [12:21.099](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=741.0988124999999): fric\_b is redrawn as the numbers it depends on change. - [12:21.099](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=741.0988124999999): sep ticks to 0.0. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil\_2 is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil\_3 is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil\_4 is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil\_5 is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil\_6 is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil\_7 is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): coil\_8 is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): block\_a is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): block\_b is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): shared\_v is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): fric\_a is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): fric\_b is redrawn as the numbers it depends on change. - [12:23.003](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=743.0028124999999): comp ticks to 0.95. - [12:25.847](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=745.8468124999999): setup is shown on the screen, written out. ##### [12:31.068](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=751.0678125) Narration: Initial kinetic energy first. The blocks are being held at rest, so T initial is zero. Board: stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)); head\_setup — a Heading that says "The Five Terms" Actions: - [12:35.12](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=755.1198125): setup is shown on the screen, written out. ##### [12:38.25](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=758.2503125) Narration: Initial potential next. Gravity contributes nothing, because nothing changes height on a level floor. The spring contributes one half k d squared, the result we derived a moment ago. Board: Unchanged from the preceding beat in this scene. Actions: - [12:45.891](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=765.8908124999999): setup is shown on the screen, written out. ##### [12:50.925](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=770.9253124999999) Narration: Final potential. At the natural length the spring is undeformed, so V final is zero as well. Board: Unchanged from the preceding beat in this scene. Actions: - [12:53.607](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=773.6068124999999): neutral is indicated — a transient flash. - [12:54.827](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=774.8268125): setup is shown on the screen, written out. ##### [12:58.514](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=778.5143125) Narration: And final kinetic energy. At that instant the blocks are still moving together with one shared speed, so it is one half the total mass, times that speed squared. Board: Unchanged from the preceding beat in this scene. Actions: - [13:3.309](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=783.3088124999999): shared\_v is indicated — a transient flash. - [13:4.145](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=784.1448125): setup is shown on the screen, written out. - [13:8.661](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=788.6613124999999): head\_setup is hidden from the screen — left the board. - [13:8.661](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=788.6613124999999): setup is hidden from the screen — left the board. ##### [13:9.261](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=789.2613124999999) Narration: Which leaves friction, and friction is not conservative: there is no potential for it, so we compute its work directly. The floor has to hold up both blocks, so the normal force is the total weight, m A plus m B, times g. Board: stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)) Actions: - [13:9.261](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=789.2613124999999): head\_friction is shown on the screen, written out. - [13:17.656](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=797.6558124999999): fric\_a is shown on the screen, written out. - [13:17.816](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=797.8159119715573): fric\_b is shown on the screen, written out. - [13:18.944](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=798.9438124999999): rough is shown on the screen, written out. ##### [13:24.2](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=804.1998124999999) Narration: Kinetic friction is mu k times that normal force, and it points leftwards the whole way, because the blocks are travelling rightwards. Board: stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)); head\_friction — a Heading that says "The Work of Friction"; fric\_a — a Vector \[magenta\] labelled "F\_f" drawn in stage (start=(((0.4 + (2.3 - comp)) + 0.4), 0.78), end=(((0.4 + (2.3 - comp)) - 0.1), 0.78)); fric\_b — a Vector \[magenta\] drawn in stage (start=(((((0.4 + (2.3 - comp)) + 0.75) + sep) + 0.35), 0.75), end=(((((0.4 + (2.3 - comp)) + 0.75) + sep) - 0.1), 0.75)) Actions: - [13:24.421](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=804.4208124999999): rough is shown on the screen, written out. - [13:28.24](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=808.2398125): fric\_a is indicated — a transient flash. ##### [13:32.706](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=812.7063125) Narration: Force against motion means negative work. It is minus the friction force times the distance travelled, which is minus mu k, times the total mass, times g d. Board: Unchanged from the preceding beat in this scene. Actions: - [13:34.529](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=814.5288125): rough is shown on the screen, written out. - [13:40.091](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=820.0908125): rough is shown on the screen, written out. ##### [13:45.021](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=825.0208124999999) Narration: Notice that this is the one place where energy saves us nothing. For a non-conservative force you always do the work integral yourself. It is simply that this particular integral is trivial: a constant force over a straight run. Board: Unchanged from the preceding beat in this scene. Actions: - [13:56.051](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=836.0508124999999): rough is indicated — a transient flash. - [13:59.394](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=839.3943125): head\_friction is hidden from the screen — left the board. - [13:59.394](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=839.3943125): rough is hidden from the screen — left the board. ##### [13:59.994](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=839.9943125) Narration: Now put the five pieces into the master equation, and double both sides to clear the halves. Zero, plus k d squared, minus twice mu k times the total weight times d, equals the total mass times the shared speed squared. Board: stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)); fric\_a — a Vector \[magenta\] labelled "F\_f" drawn in stage (start=(((0.4 + (2.3 - comp)) + 0.4), 0.78), end=(((0.4 + (2.3 - comp)) - 0.1), 0.78)); fric\_b — a Vector \[magenta\] drawn in stage (start=(((((0.4 + (2.3 - comp)) + 0.75) + sep) + 0.35), 0.75), end=(((((0.4 + (2.3 - comp)) + 0.75) + sep) - 0.1), 0.75)) Actions: - [13:59.994](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=839.9943125): head\_solve is shown on the screen, written out. - [14:6.16](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=846.1598124999999): solve is shown on the screen, written out. ##### [14:17.139](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=857.1388124999999) Narration: Solve for the speed squared. Divide both sides by the total mass, and there it is: k d squared, minus twice mu k times total mass times g d, all over the total mass. Board: stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)); fric\_a — a Vector \[magenta\] labelled "F\_f" drawn in stage (start=(((0.4 + (2.3 - comp)) + 0.4), 0.78), end=(((0.4 + (2.3 - comp)) - 0.1), 0.78)); fric\_b — a Vector \[magenta\] drawn in stage (start=(((((0.4 + (2.3 - comp)) + 0.75) + sep) + 0.35), 0.75), end=(((((0.4 + (2.3 - comp)) + 0.75) + sep) - 0.1), 0.75)); head\_solve — a Heading that says "Solving for the Condition" Actions: - [14:17.487](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=857.4868124999999): solve is shown on the screen, written out. ##### [14:32.588](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=872.5878124999999) Narration: B separates only if that speed is genuinely positive. The denominator is positive already, so the whole condition sits in the numerator: k d squared has to beat twice mu k, total mass, g, d. Board: Unchanged from the preceding beat in this scene. Actions: - [14:40.715](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=880.7148125): solve (the "k d^2 - 2 mu\_k (m\_A + m\_B) g d" part) is emphasized. - [14:43.397](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=883.3968124999999): solve is shown on the screen, written out. - [14:47.971](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=887.9713125): solve (the "k d^2 - 2 mu\_k (m\_A + m\_B) g d" part) is no longer emphasized. ##### [14:48.571](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=888.5713125) Narration: Every compression is positive, so divide one factor of d out of both sides. The answer: d must be bigger than twice mu k, total mass, g, over k. Board: Unchanged from the preceding beat in this scene. Actions: - [14:54.098](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=894.0978124999999): solve is shown on the screen, written out. - [14:55.607](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=895.6068124999999): A box is drawn around solve. ##### [14:59.957](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=899.9573125) Narration: Compare that with the route through Newton's laws. There you write F equals m a for the pair, with a spring force that changes with position, integrate to get speed against position, and only then set the speed to zero. Same answer, several times the work. Board: Unchanged from the preceding beat in this scene. Actions: - [15:0.178](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=900.1778125000001): compare is shown on the screen, written out. ##### [15:16.416](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=916.4158124999999) Narration: And that is the shape of every energy problem. Pick the two states, write T plus V at each of them, add the work of anything non-conservative, and solve. Almost all of the mechanics is in choosing the two states well. Board: stage — a Figure (x\_range=(0.0, 6.6), y\_range=(0.0, 2.6), aspect=(6.6, 2.6)); wall — a Line \[gray\] drawn in stage (start=(0.4, 0.55), end=(0.4, 1.85)); bench — a Line \[gray\] drawn in stage (start=(0.15, 0.55), end=(6.45, 0.55)); mu\_mark — a Math \[gray\] that says "$mu\_k$" drawn in stage; coil — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.0)), 0.95), end=((0.4 + ((2.3 - comp) \* 0.125)), 1.17)); coil\_2 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.125)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.25)), 0.73)); coil\_3 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.25)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.375)), 1.17)); coil\_4 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.375)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.5)), 0.73)); coil\_5 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.5)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.625)), 1.17)); coil\_6 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.625)), 1.17), end=((0.4 + ((2.3 - comp) \* 0.75)), 0.73)); coil\_7 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.75)), 0.73), end=((0.4 + ((2.3 - comp) \* 0.875)), 1.17)); coil\_8 — a Line \[yellow\] drawn in stage (start=((0.4 + ((2.3 - comp) \* 0.875)), 1.17), end=((0.4 + ((2.3 - comp) \* 1.0)), 0.95)); block\_a — a Polygon \[blue\] labelled "A" drawn in stage (vertices=(((0.4 + (2.3 - comp)), 0.55), (((0.4 + (2.3 - comp)) + 0.75), …, fill\_opacity=0.45); block\_b — a Polygon \[green\] labelled "B" drawn in stage (vertices=(((((0.4 + (2.3 - comp)) + 0.75) + sep), 0.55), (((((0.4 + (2.3…, fill\_opacity=0.45); neutral — a Line \[gray\] drawn in stage (start=(2.6999999999999997, 0.55), end=(2.6999999999999997, 2.0), dashed=True); d\_mark — a Line \[yellow\] labelled "d" drawn in stage (start=(1.7499999999999998, 1.95), end=(2.6999999999999997, 1.95)); shared\_v — a Vector \[green\] labelled "arrow(v)" drawn in stage (start=((((0.4 + (2.3 - comp)) + 0.75) + 0.05), 1.62), end=((((0.4 + (2.3 - comp)) + 0.75) + 0.85), 1.62)); fric\_a — a Vector \[magenta\] labelled "F\_f" drawn in stage (start=(((0.4 + (2.3 - comp)) + 0.4), 0.78), end=(((0.4 + (2.3 - comp)) - 0.1), 0.78)); fric\_b — a Vector \[magenta\] drawn in stage (start=(((((0.4 + (2.3 - comp)) + 0.75) + sep) + 0.35), 0.75), end=(((((0.4 + (2.3 - comp)) + 0.75) + sep) - 0.1), 0.75)); compare — a Block \[text\] that says "Energy: two states, one equation. $F = m a$: integrate first."; head\_solve — a Heading that says "Solving for the Condition" Actions: - [15:19.992](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=919.9918125): point\_3 is shown on the screen, grown. - [15:22.343](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=922.342572234093): point\_3 is hidden from the screen. - [15:22.801](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=922.800822629788): point\_4 is shown on the screen, grown. - [15:24.574](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=924.5737162142556): point\_4 is hidden from the screen. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): compare is hidden from the screen — left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): head\_solve is hidden from the screen — left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): solve is hidden from the screen — left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): stage is hidden from the screen — left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): wall is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): bench is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): mu\_mark is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil\_2 is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil\_3 is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil\_4 is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil\_5 is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil\_6 is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil\_7 is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): coil\_8 is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): block\_a is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): block\_b is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): neutral is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): d\_mark is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): shared\_v is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): fric\_a is hidden from the screen — stage left the board. - [15:30.129](https://academa.ai/@atalay/lectures/the-mechanics-and-intuition-of-potential-energy?t=930.129375): fric\_b is hidden from the screen — stage left the board.